Closure property of consistently varying random variables based on precise large deviation principles

Shijie Wang, Duo Guo and Wensheng Wang

Communications in Statistics - Theory and Methods, 2019, vol. 48, issue 9, 2218-2228

Abstract: Let X = {X1, X2, …} be a sequence of independent but not necessarily identically distributed random variables, and let η be a counting random variable independent of X. Consider randomly stopped sum Sη = ∑ηk = 1Xk and random maximum S(η) ≕ max {S0, …, Sη}. Assuming that each Xk belongs to the class of consistently varying distributions, on the basis of the well-known precise large deviation principles, we prove that the distributions of Sη and S(η) belong to the same class under some mild conditions. Our approach is new and the obtained results are further studies of Kizinevič, Sprindys, and Šiaulys (2016) and Andrulytė, Manstavičius, and Šiaulys (2017).

Date: 2019
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DOI: 10.1080/03610926.2018.1459717

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